Quadratic Functions: Concepts, Graphing, and Solution Methods

Quadratic Functions

Definition and General Form

A quadratic function is a polynomial function of degree two, typically written in the form:

• Standard Form: $f(x) = ax^2 + bx + c$, where a, b, and c are real numbers and a ≠ 0.

• The graph of a quadratic function is a parabola.

Standard Form of a Quadratic Function

The standard form can also be expressed as:

• $f(x) = a(x - h)^2 + k$

• Vertex: $(h, k)$

• The parabola is symmetric about the vertical line $x = h$ (called the axis of symmetry).

• If $a > 0$, the parabola opens upward and $k$ is the minimum value.

• If $a < 0$, the parabola opens downward and $k$ is the maximum value.

Graphical Features of Quadratic Functions

Key features of the graph include:

• Vertex: The highest or lowest point of the parabola.

• Axis of Symmetry: The vertical line passing through the vertex, $x = h$.

• Direction: Determined by the sign of $a$ (upward for $a > 0$, downward for $a < 0$).

How to Graph a Quadratic Function

To graph $f(x) = a(x - h)^2 + k$ or $f(x) = ax^2 + bx + c$:

1. Identify $a$, $h$, and $k$.

2. Determine if the parabola opens upward ($a > 0$) or downward ($a < 0$).

3. Find the vertex $(h, k)$.

4. Find the axis of symmetry ($x = h$).

5. Find $y$-intercept by setting $x = 0$.

6. Find $x$-intercepts (roots) by solving $f(x) = 0$.

7. Plot the vertex, intercepts, and additional points as needed.

Finding the Vertex from General Form

For $f(x) = ax^2 + bx + c$:

• The vertex's $x$-coordinate is $x = -\frac{b}{2a}$.

• The $y$-coordinate is $k = f(-\frac{b}{2a})$.

Examples

• Example 1: Writing the Equation of a Quadratic Function Find the standard form of a quadratic function whose graph has vertex $(3,4)$ and passes through the point $(4,7)$. Does it have a maximum or minimum value? Solution: Use $f(x) = a(x - 3)^2 + 4$ and substitute $(4,7)$ to solve for $a$.

• Example 2: Graphing a Quadratic Function in Standard Form Sketch the graph of $f(x) = -3(x - 2)^2 + 12$. State the vertex and axis of symmetry. Solution: Vertex: $(2,12)$, Axis of symmetry: $x = 2$, Opens downward ($a < 0$).

• Example 3: Graphing a Quadratic Function Sketch the graph of $f(x) = 2x^2 + 8x - 10$.

• Example 4: Identifying Characteristics from a Graph Given $f(x) = 2x^2 + 8x - 5$, find the vertex and axis of symmetry.

Solving Quadratic Equations

Quadratic equations can be solved by several methods:

• Factoring: Set $f(x) = 0$ and factor the quadratic expression.

• Square Root Method: For equations of the form $x^2 = k$, solve by taking square roots.

• Quadratic Formula: For $ax^2 + bx + c = 0$, use: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Review Examples

• Example 5: Solve by Factoring $x^2 - 14 = 0$ Solution: $x^2 = 14 \Rightarrow x = \pm \sqrt{14}$

• Example 6: Solve by Square Root Method $(x+1)^2 = 16$ Solution: $x+1 = \pm 4 \Rightarrow x = 3$ or $x = -5$

• Example 7: Solve by Quadratic Formula $x^2 + 2x + 4 = 0$ Solution: $x = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 1 \cdot 4}}{2 \cdot 1} = \frac{-2 \pm \sqrt{-12}}{2}$

Summary Table: Methods for Solving Quadratic Equations

Method	When to Use	Example
Factoring	When the quadratic can be factored easily	$x^2 - 14 = 0$
Square Root	When the equation is in the form $(x + d)^2 = k$	$(x+1)^2 = 16$
Quadratic Formula	For any quadratic equation	$x^2 + 2x + 4 = 0$

Additional info: The notes also briefly mention the importance of the axis of symmetry and the vertex in graphing and analyzing quadratic functions. The examples provided cover both graphing and solving quadratic equations using multiple methods.